![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > gictr | Structured version Visualization version GIF version |
Description: Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
gictr | ⊢ ((𝑅 ≃𝑔 𝑆 ∧ 𝑆 ≃𝑔 𝑇) → 𝑅 ≃𝑔 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brgic 18192 | . 2 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) | |
2 | brgic 18192 | . 2 ⊢ (𝑆 ≃𝑔 𝑇 ↔ (𝑆 GrpIso 𝑇) ≠ ∅) | |
3 | n0 4190 | . . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆)) | |
4 | n0 4190 | . . 3 ⊢ ((𝑆 GrpIso 𝑇) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑆 GrpIso 𝑇)) | |
5 | exdistrv 1915 | . . . 4 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑔 ∈ (𝑆 GrpIso 𝑇)) ↔ (∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ ∃𝑔 𝑔 ∈ (𝑆 GrpIso 𝑇))) | |
6 | gimco 18191 | . . . . . . 7 ⊢ ((𝑔 ∈ (𝑆 GrpIso 𝑇) ∧ 𝑓 ∈ (𝑅 GrpIso 𝑆)) → (𝑔 ∘ 𝑓) ∈ (𝑅 GrpIso 𝑇)) | |
7 | brgici 18193 | . . . . . . 7 ⊢ ((𝑔 ∘ 𝑓) ∈ (𝑅 GrpIso 𝑇) → 𝑅 ≃𝑔 𝑇) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ ((𝑔 ∈ (𝑆 GrpIso 𝑇) ∧ 𝑓 ∈ (𝑅 GrpIso 𝑆)) → 𝑅 ≃𝑔 𝑇) |
9 | 8 | ancoms 451 | . . . . 5 ⊢ ((𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑔 ∈ (𝑆 GrpIso 𝑇)) → 𝑅 ≃𝑔 𝑇) |
10 | 9 | exlimivv 1892 | . . . 4 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑔 ∈ (𝑆 GrpIso 𝑇)) → 𝑅 ≃𝑔 𝑇) |
11 | 5, 10 | sylbir 227 | . . 3 ⊢ ((∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ ∃𝑔 𝑔 ∈ (𝑆 GrpIso 𝑇)) → 𝑅 ≃𝑔 𝑇) |
12 | 3, 4, 11 | syl2anb 589 | . 2 ⊢ (((𝑅 GrpIso 𝑆) ≠ ∅ ∧ (𝑆 GrpIso 𝑇) ≠ ∅) → 𝑅 ≃𝑔 𝑇) |
13 | 1, 2, 12 | syl2anb 589 | 1 ⊢ ((𝑅 ≃𝑔 𝑆 ∧ 𝑆 ≃𝑔 𝑇) → 𝑅 ≃𝑔 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∃wex 1743 ∈ wcel 2051 ≠ wne 2960 ∅c0 4172 class class class wbr 4925 ∘ ccom 5407 (class class class)co 6974 GrpIso cgim 18180 ≃𝑔 cgic 18181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-1st 7499 df-2nd 7500 df-1o 7903 df-map 8206 df-0g 16569 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-grp 17906 df-ghm 18139 df-gim 18182 df-gic 18183 |
This theorem is referenced by: gicer 18199 cyggic 20436 |
Copyright terms: Public domain | W3C validator |